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The triangle medians and the centroid.. In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. . Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent
For instance, if three cevians of a triangle intersect at a point P, then their isogonal lines also intersect at a point, called the isogonal conjugate of P. The symmedians illustrate this fact. In the diagram, the medians (in black) intersect at the centroid G .
Three of them are the medians, which are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given ...
There is only one automedian right triangle, the triangle with side lengths proportional to 1, the square root of 2, and the square root of 3. [2] This triangle is the second triangle in the spiral of Theodorus. It is the only right triangle in which two of the medians are perpendicular to each other. [2]
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. [1] [2] Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name. [3]
Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid. Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter.
The medians coincide at the triangle's centroid, which is also the center of the Steiner ellipse. Not to be confused with Steiner conic . In geometry , the Steiner ellipse of a triangle is the unique circumellipse (an ellipse that touches the triangle at its vertices ) whose center is the triangle's centroid . [ 1 ]
For 3 (non-collinear) points, if any angle of the triangle formed by those points is 120° or more, then the geometric median is the point at the vertex of that angle. If all the angles are less than 120°, the geometric median is the point inside the triangle which subtends an angle of 120° to each three pairs of triangle vertices. [ 10 ]